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Hazard found enough space after a swift passing move to rifle home from 20 metres and score against the run of play in the 42nd minute, setting up a quarterfinal clash with Italy.
The loss of De Ligt was immediately felt as the Dutch went from being comfortably in control to being forced onto the back foot by a muscular Czech side.
Striker Patrik Schick scored his fourth goal of the tournament as the Czech Republic upset 10-man Netherlands with a deserved 2-0 victory.
Perisic, who scored the equaliser in a 1-1 draw with the Czech Republic and a late goal in a 3-1 win over Scotland in the group stage, will be left behind in self-isolation as Croatia travel to Copenhagen for the clash with the Spaniards on Monday.
Italy's squad depth proved decisive in the end as Juventus winger Chiesa and Atalanta midfielder Pessina spared their blushes with superb finishes.
Forward Kasper Dolberg struck in each half while Joakim Maehle and Martin Braithwaite also got in on the act late on as Denmark overpowered Wales 4-0.
The result ensures that with one round of group games still to play, Paraguay qualify from Group A along with Chile, Argentina and Uruguay.
Nestor Pitana, the Argentine who refereed the 2018 World Cup final, waved play on when the ball bounced off him outside Colombia's box and Brazil continued passing for Roberto Firmino to head in an equaliser.
Games now tied on aggregate score after the regulation 90 minutes in the second leg will go direct to extra time and then to a penalty shootout.
The Danish national team has long been influenced by the Dutch masters at Ajax Amsterdam, and plenty of Danish players like Michael Laudrup, Morten Olsen, as a manager, and current striker Kasper Dolberg have thrived in the Dutch capital.
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